We now consider an analytically solvable model for the flow. Let us assume that the network consists of a set of parallel links placed
between two fluid reservoirs kept at pressure \(p=0\)  and \(p=\nabla p<0\) , i.e., we are describing the capillary fiber bundle model
[1]}, [2]}, [3]}. The constitutive equation for the fiber bundle is then given by
\(Q&=&\sum _{i=1}^{N_y} \left[-\Theta (q_i-\alpha |\nabla p|) \alpha \nabla p\right.\nonumber \\&-&\Theta (\alpha |\nabla p|-q_i)\beta \nabla p\nonumber \\&+&\left.\Theta (\alpha |\nabla p|-q_i)\left(1-\frac{\beta }{\alpha }\right)q_i\right]\;,\) 
